Multiple solutions for p-Laplacian type problems with asymptotically p-linear terms via a cohomological index theory

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The aim of this paper is investigating the existence of weak solutions of the quasilinear elliptic model problem {arraylr - (A(x,u) |∇ u|ᵖ⁻² ∇ u) + 1p Aₜ(x,u) |∇ u|ᵖ=f(x,u) & in Ω, u= 0 & on ∂Ω, array. where Ω⊂ ⁿ is a bounded domain, N≥ 2, p > 1, A is a given function which admits partial derivative Aₜ(x,t) = ∂ A/∂ t(x,t) and f is asymptotically p-linear at infinity. Under suitable hypotheses both at the origin and at infinity, and if A(x,·) is even while f(x,·) is odd, by using variational tools, a cohomological index theory and a related pseudo--index argument, we prove a multiplicity result if p > N in the non--resonant case.